Change in Linear and Exponential Functions - AP Precalculus

1. What is the effect of a zero slope in a linear function?

   Horizontal line. Zero slope creates constant function value.

2. If $f(x) = 6x - 9$, what is $f(2)$?

   3. Substitute $x = 2$: $f(2) = 6(2) - 9 = 3$.

3. Find the slope of the line passing through points $ (1, 2) $ and $ (3, 6) $.

   2. Slope = $\frac{6-2}{3-1} = \frac{4}{2} = 2$.

4. Find the slope of the line passing through points $(1, 2)$ and $(3, 6)$.

   2. Slope = $\frac{6-2}{3-1} = \frac{4}{2} = 2$.

5. Find $f(x)$ if $f(x) = 2 \times 3^x$ and $x = 2$.

   18. Substitute $x = 2$: $f(2) = 2 \times 3^2 = 18$.

6. What is the formula for an exponential function?

   $f(x) = a \times b^x$. Where $a$ is initial value and $b$ is the base.

7. Identify the linear function: $f(x) = 2x - 5$ or $f(x) = 3^x$?

   $f(x) = 2x - 5$. Form $mx + b$ indicates linear function.

8. What does a negative exponent in an exponential function represent?

   Reciprocal growth. Negative exponent means division by positive power.

9. Find the rate of change for $f(x) = 5x + 3$.

   $5$. The coefficient of $x$ is the rate of change.

10. What is the rate of change in $f(x) = x + 4$?

    1. The slope is the coefficient of $x$.

11. Describe the graph of $f(x) = b^x$ when $0 < b < 1$.

    Decaying exponential. Base less than 1 creates decreasing exponential.

12. What is the general form of a linear function?

    $f(x) = mx + b$. Standard form with slope $m$ and y-intercept $b$.

13. Describe the graph of $f(x) = b^x$ when $b > 1$.

    Growing exponential. Base greater than 1 creates increasing exponential.

14. Find the initial value of $f(x) = 10 \times 0.9^x$.

    10. The coefficient multiplying the exponential term.

15. What is the impact of a negative slope in a linear function?

    Decreasing function. Negative slope means function decreases as $x$ increases.

16. What is the common ratio in $f(x) = 7 \times 3^x$?

    3. The base of the exponential term.

17. Find the y-intercept of $f(x) = -3x + 12$.

    12. Y-intercept occurs when $x = 0$.

18. State the decay factor in $f(x) = 5 \times 0.8^x$.

    0.8. The base represents decay when between 0 and 1.

19. State the initial value in $f(x) = 4 \times 2^x$.

    4. The coefficient multiplying the exponential term.

20. Identify the y-intercept in the linear function $f(x) = -2x + 5$.

    5. The constant term is the y-intercept.

21. If $f(x) = 4 \times 2^x$, what is $f(3)$?

    32. Substitute $x = 3$: $f(3) = 4 \times 2^3 = 32$.

22. What is the effect of doubling the coefficient in a linear function?

    Steeper slope. Larger coefficient creates steeper line.

23. What does the y-intercept represent in a linear function?

    Initial value. Y-intercept is the starting value when $x = 0$.

24. What happens when the exponent in an exponential function is negative?

    Function decreases. Negative exponent creates reciprocal, decreasing function.

25. Find $f(0)$ for $f(x) = 9 \times 2^x$.

    9. At $x = 0$: $f(0) = 9 \times 2^0 = 9$.

26. Determine the type: $f(x) = 4 \times 1.5^x$.

    Exponential function. Form $a \times b^x$ identifies exponential function.

27. Calculate $f(0)$ for $f(x) = 5x - 7$.

    -7. At $x = 0$: $f(0) = 5(0) - 7 = -7$.

28. Identify the growth factor in $f(x) = 2 \times 1.1^x$.

    1.1. The base represents growth when greater than 1.

29. Which function type has a constant ratio between outputs?

    Exponential function. Exponential functions have constant multiplicative rate.

30. Identify the exponential function: $f(x) = 5x$ or $f(x) = 2 \times 3^x$?

    $f(x) = 2 \times 3^x$. Form $a \times b^x$ indicates exponential function.

31. Which function type has a constant rate of change?

    Linear function. Linear functions have constant slope.

32. Identify the slope in the linear function $f(x) = 3x + 7$.

    3. The coefficient of $x$ is the slope.

33. What is the y-intercept of $f(x) = 6x - 8$?

    -8. The constant term when $x = 0$.

34. Find $f(4)$ for $f(x) = 7 \times 2^x$.

    112. Substitute $x = 4$: $f(4) = 7 \times 2^4 = 112$.

35. Determine the type: $f(x) = 0.5x - 2$.

    Linear function. Form $mx + b$ identifies linear function.

36. What is the base in the exponential function $f(x) = 3 \times (0.5)^x$?

    0.5. The number being raised to the power.

37. What is the effect of doubling the base in an exponential function?

    Increases the growth rate. Larger base means faster exponential growth.

38. What does the slope represent in a linear function?

    Rate of change. Slope shows how much $y$ changes per unit of $x$.

39. Calculate $f(1)$ for $f(x) = 3 \times (0.5)^x$.

    1.5. At $x = 1$: $f(1) = 3 \times (0.5)^1 = 1.5$.

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